The nature of glass transitions in chalcogenides and modified oxides depends on the network mean coordination number $\langle r\rangle$. These display systematic trends when spanning across the three topological phases: flexible, intermediate, and stressed-rigid. Trends in the glass-transition temperature Tg($\langle r\rangle$) show a monotonic increase with $\langle r\rangle$, but the nonreversing enthalpy of relaxation at Tg, ΔHnr($\langle r\rangle$), shows a deep- and square-well-like minimum with the walls representing the rigidity and stress transitions with increasing $\langle r\rangle$, respectively. In the well, the ΔHnr($\langle r\rangle$) term remains minuscule (∼0) corresponding to the isostatically rigid intermediate phase (IP). The melt fragility index (m) shows rather low values, m($\langle r\rangle$) < 20 for IP compositions, but increases outside the IP. Glass compositions in the IP show absence of network stress, form compacted networks, possess thermally reversing glass transitions, and display high glass-forming tendency—functionalities that have attracted widespread interest in understanding the physics of glasses and applications of the new IP formed.

We investigate the asymptotic distribution of the number of exceedances among d identically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of the fragility index, which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the first d RVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence as d increases.